Environmental Engineering Reference
In-Depth Information
6.9.2 The Perturbation Method for Solving
Hyperbolic Heat-Conduction Equations
Consider
P
0
:
u
t
A
2
u
xx
R
1
/
τ
+
=
,
×
(
,
+
∞
)
,
u
tt
0
0
(6.169)
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
P
N
(
x
)
.
1. When
N
is even such that
N
=
2
m
, the solution of PDS (6.169) reads, by
Eq. (6.168),
u
2
N
n
d
u
I
0
b
2τ
0
+
At
−
1
t
2
A
e
−
n
=
0
a
n
(
u
+
x
)
u
=
(
At
)
2
−
(6.170)
At
u
2
N
n
=
0
a
n
n
k
=
0
C
n
u
k
x
n
−
k
d
u
.
I
0
b
τ
0
+
At
−
1
t
2
A
e
−
2
=
2
(
At
)
−
(6.171)
At
For convenience, define
I
0
b
u
2
u
i
d
u
+
At
G
i
(
t
)=
(
At
)
2
−
,
i
=
0
,
1
, ···
n
,
(6.172)
−
At
where
I
0
(
x
)
is the modified Bessel function of order zero and the first kind. Since
I
0
(
is defined by a power series that converges very quickly, we can easily
obtain
G
i
(
x
)
t
)
by integration term by term. The
G
i
(
t
)
can sometimes be expressed
by elementary functions; for example,
I
0
b
u
2
d
u
+
At
τ
0
1
τ
0
t
t
0
e
−
e
−
(
)=
(
)
2
−
=
−
.
G
0
t
At
2
A
τ
2
(6.173)
−
At
(
)=
When
i
is odd,
G
i
t
0.
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